1

Chapter 6

It is a truth very certain that when it is not in our power to determine what is true we ought to follow what is probable.—Descartes

Decision Making and Risk:Certainty Equivalents

and Utility

2

Decision Making Using Certainty Equivalents

The certainty equivalent (CE) is the payoff amount we would accept in lieu of under- going the uncertain situation.

Shirley Smart would pay $25 to insure her 1983 Toyota against total theft loss. CE = – $25.

For $1,000, Willy B. Rich would sell his Far- Fetched Lottery rights. CE = $1,000.

Lucky Chance would pay somebody $100 to fill her Far-Fetched Lottery contract. CE = $100.

3

Risk Premiums

A situation’s risk premium (RP) is the difference its between expected payoff (EP) and certainty equivalent (CE):

RP = EP CE Shirley Smart’s car is worth $1,000 and

there is a 1% chance of its being stolen. Thus, going without insurance has

EP = (– $1,000)(.01) + ($0)(.99) = – $10

RP = – $10 – (– $25) = $15

4

Risk Premiums

Playing the Far-Fetched Lottery has EP = $2,500. Thus, For Willy B. Rich,

RP = EP – CE = $2,5000 – ($1,000) = $1,500 For Lucky Chance,

RP = EP – CE = $2,5000 – (– $100) = $2,600

Different people will have different CEs and RPs for the same circumstance. They have different attitudes toward risk.

5

Attitude Toward Risk

People with positive RPs are risk averters. Lucky Chance has greater risk aversion than

Willy B. Rich, as reflected by her greater RP. We cannot compare Shirley’s risk aversion to

the others’ because circumstances differ.

Risk averse persons have RPs that increase: When the downside amounts become greater. Or when the chance of downside increases.

A risk seeker will have negative RP. A risk neutral person has zero RP.

6

Maximizing Certainty Equivalent

A plausible axiom:Decision makers will prefer the act yielding greatest certainty equivalent.

A logical conclusion:The ideal decision criterion is to maximizecertainty equivalent. Doing so guarantees taking the preferred action.

But CEs are difficult to determine. One approach is to discount the EPs. RP = EP – CE implies that CE = EP – RP.

7

Using Risk Premiums to Get Certainty Equivalents

Ponderosa Records president has the following risk premiums, keyed to the downside.

These were found by extrapolating from three equivalencies (white boxes). Exact amounts are unknowable, but these values seem

to fit his risk profile.

8

Decision Tree Analysis with CEs (Discounted Expected Payoffs)

9

How Good is the Analysis?

This result is different from that of ordinary back folding (Bayes decision rule). It specifically reflects underlying risk aversion. The result must be correct if CEs are right.

The major weakness is the ad hoc manner for getting the RPs, and hence the CEs. Many assumptions are made in extrapolating to

get the table of RPs.

There is a cleaner way to achieve the same thing using utilities.

10

Utility Theory

Consider a set of outcomes, O1, O2, ..., On. The following assumptions are made: Preference ranking can be done. Transitivity of preference: A is preferred to B

and B to C, then A must be preferred to C. Continuity: Consider Obetween. Take a gamble

between two more extreme outcomes; winning yields Obest and losing Oworst. There is a win probability q making you indifferent between

getting Obetween and gambling. Such a gamble is called a reference lottery.

11

Utility Assumptions

Continuity (continued): e.g., +$1,000 v. Far-Fetched Lottery, you pick q. For Willy B. Rich, q = .5. (His CE was = +$1,000.) For Lucky Chance, q = .9. If the win probability were .99, would you risk +

$1,000 to gamble? What is your q?

Substitutability: In a decision structure, you would willingly substitute for any outcome a gamble equally preferred. One outcome on Lucky Chance’s tree is +$1,000;

she would accept substituting for it the Far-Fetched Lottery gamble with .9 win probability.

12

Utility Assumptions and Values

Increasing preference: Raising q makes any reference lottery more preferred. Anybody would prefer the revised Far-Fetched

Lottery when two coins are tossed and just one head will win the $10,000. (The win probability goes from .5 to .75.) You still might not like that gamble!

Outcomes can be assigned utility values arbitrarily, so that the more preferred always gets the greater value:

u(Obest) = 10 u(Oworst)=0 u(Obetween)=50 Willy has u(+$10,000) = 500, u$5,000) = 0 and u(+

$1,000) = 250. These are his values only.

13

Utility Values Lucky has different values: u(+$10,000) = 50,

u$5,000) = 99, and u(+$1,000) = 35.1. Like temperature, where 0o and 100o are

different states on Celsius and Fahrenheit scales, so utility scales may differ.

The freezing point for water is 0o C and the boiling point 100o C. In-between states will have values in that range, and hotter days will have greater temperature values than cooler.

So, too, with utility values. They will fall into the range defined by the extreme outcomes, Oworst and Obest. More preferred outcomes will have greater utilities

14

Utility Values

A reference lottery can be used to find the utility for an outcome Obetween by: First, establish an indifference win probability

qbetween making it equally preferred to the gamble: Obest with probability qbetween and

Oworst with probability 1 qbetween

Second, compute the lottery’s expected utility: u(Obest)(qbetween) + u(Oworst)(1 qbetween)

The above is u(Obetween).

15

Utility Values

Using the Far-Fetched Lottery as reference:

The indifference q plays a role analogous to the thermometer, reading attitude towards the outcome similarly to measuring temperature.

Lottery

Outcomes

Willy Lucky

Prob. Utility Prob. Utility

Obest (+$10,000) q=.5 500 q=.9 50

Oworst ($5,000) 1 .5 0 1 .9 99

Expected Utility: 250 35.1

Obetween (+$1,000): 250 35.1

16

The Utility Function

Utility values assigned to monetary outcomes constitute a utility function.

From a few points we may graph the utility function and apply it over a monetary range.

Those points may be obtained from an interview posing hypothetical gambles. Using u(+$10,000)=100 and u$5,000)=0

Shirley Smart gave the following equivalencies: A: +$10,000 @ qA v $5,000 ≡ +$1,000 if qA =.70 B: +$10,000 @ qB v +$1,000 ≡ +$5,000 if qB =.75 C1: +$1,000 @ qC1 v $5,000 ≡ $500 if qC1 =.70 C2: +$1,000 @ qC2 v $5,000 ≡ $2,000 if qC2 =.30

17

Shirley’s Utility Function

Shirley’s utilities for the equivalent amounts are equal to the respective expected utilities: u(+$1,000) = u(+$10,000)(.70) + u$1,000)(1.70)

= 100(.70) + 0(1.70) = 70 u(+$5,000) = u(+$10,000)(.75) + u$1,000)(1.75)

= 100(.75) + 70(1 .75) = 92.5 u$500) = u(+$1,000)(.70) + u$5,000)(1.70)

= 70(.70) + 0(1 .70) = 49 u($2,000) = u(+$1,000)(.30) + u$5,000)(1.30)

= 70(.30) + 0(1 .30) = 21 Altogether, Shirley gave 6 points, plotted on the

following graph. The smoothed curve fitting through them defines her utility function.

18

Shirley’s Utility Function

19

Using the Utility Function

This utility function applies to the Ponderosa decision.

20

Using the Utility Function

Read the utility payoffs corresponding to the net monetary payoffs.

Apply the Bayes decision rule, with either: A utility payoff table, computing the expected

payoff each act. Or a decision tree, folding it back.

The certainty equivalent amount for any act or node may be found from the expected utility by reading the curve in reverse.

The following Ponderosa Records decision tree was folded back using utility payoffs.

21

Decision Tree Analysiswith Utilities

22

Shape of Utility Curve and Attitude Toward Risk

The following shapes generally apply.

The risk averter has decreasing marginal util- ility for money. He will buy casualty insurance and losses weigh more heavily than like gains.

Risk seekers like some unfavorable gambles. Risk neutrality values money at its face amount.

23

Important Utility Ramifications

Hybrid shapes (like Shirley’s) imply shifting attitudes as monetary ranges change.

Regardless of shape, maximizing expected utility also maximizes certainty equivalent. Therefore, applying Bayes decision rule with

utility payoffs discloses the preferred action.

Primary impediments to implementation: Clumsiness of the interview process. Multiple decision makers. Attitudes change with circumstances and time.

24

Ratification ofBayes Decision Rule

Over narrow monetary ranges, utility curves resemble straight lines.

For a straight line, expected utility equals the utility of the expected monetary payoff.

Maximizing expected monetary payoff then also maximizes expected utility. Thus: The Bayes decision rule discloses the preferred

action as long as the outcomes are not extreme. Managers can then delegate decision

making without having to find utilities. Preferred actions will be found by the staff.

25

Using Utility Functions with PrecisionTree

PrecisionTree can be used to evaluate decision trees with with exponential and logarithmic utility functions.

To get started, click on the name box of a decision tree and the Tree Setting dialog box appears, as shown next.

12

A B1

0tree #1

26

Tree Settings Dialog Box (Figure 6-14)

1. Check the Use Utility Function box.

1. Check the Use Utility Function box.

2. Select the type of utility function in the Function line. Here exponential is chosen.

2. Select the type of utility function in the Function line. Here exponential is chosen.

3. Select the risk coefficient, R, in the R value line. Here 10,000 is used.

3. Select the risk coefficient, R, in the R value line. Here 10,000 is used.

4. Select Expected Utility in the Display line. Other options are Certainty Equivalent and Expected Value.

4. Select Expected Utility in the Display line. Other options are Certainty Equivalent and Expected Value.

5. Click OK.5. Click OK.

27

Decision Tree with Exponential Utility

Function for R = 10,000 (Figure 6-15) 123456789

10111213141516171819202122232425262728293031323334

A B C D E F80.0% 0.0

$90,000 1FALSE

-$50,000 -4820.0% 0.0

$0 -24450.0%

$10,000 -1TRUE 0

$0 -1FALSE

-$15,000 -220.0% 0.0

$90,000 1FALSE

-$50,000 -53180.0% 0.0

$0 -66450.0%

$0 -3TRUE 0

$0 -3

050.0% 0

$100,000 1FALSE

-$60,000 -20150.0% 0

$0 -402TRUE

$0 0TRUE 1

$0 0

Ponderosa Record Company

Test market

Don't test market

Favorable

Unfavorable

Market nationally

Abort

Success

Failure

Market nationally

Abort

Success

Failure

Market nationally

Abort

Success

Failure

The optimal strategy is:

1. Not test market and to abort.

2. The corresponding expected utility is 0.

The optimal strategy is:

1. Not test market and to abort.

2. The corresponding expected utility is 0.